Factors and Multiples: What Is the Difference?

One relationship, read from two ends

3 is a factor of 12. 12 is a multiple of 3. Those two sentences describe the same single fact - that 3 divides into 12 exactly - from opposite ends of it. Which word you need depends only on which number the sentence is about, not on anything to do with 3 or 12 themselves.

That is precisely why they are so easy to swap. There is no property of the numbers to remember. The only thing to hold on to is which one the question has put you standing on, and under exam pressure that is the thing that slips.

• Factors of 12: 1, 2, 3, 4, 6, 12 - a finite list, none of them bigger than 12
• Multiples of 12: 12, 24, 36, 48, 60, and so on - an endless list, none of them smaller than 12
• 12 appears in both lists, because every number is a factor of itself and a multiple of itself

The test that survives exam pressure

Say the sentence out loud with a plainer verb in it. For factor, ask: does it go into it? For multiple, ask: is it in that times table? Both questions have obvious answers, and neither requires you to remember which word is which.

There is a size check behind it that is faster still. Every factor of a positive number is smaller than or equal to that number, and every multiple is larger than or equal to it. So if the answer you are about to write is bigger than the number you started from, the word must be multiple. The two ranges overlap at exactly one point, where the number meets itself.

• Is 4 a factor of 20? Does 4 go into 20 exactly? Yes.
• Is 4 a multiple of 20? Is 4 in the 20 times table? No.
• Is 60 a multiple of 15? Is 60 in the 15 times table? Yes.
• Is 60 a factor of 15? Does 60 go into 15? No.

Listing factors without missing one

Factors arrive in pairs, and writing them down as pairs is what stops one going missing. Start at 1 and work upwards, recording both members of each pair, and stop when the two sides meet in the middle.

For 36 that gives 1, 2, 3, 4, 6, 9, 12, 18 and 36 - nine factors. The count is odd because 6 × 6 contributes only one number instead of two, and that is true of every square number and of no other number, which makes it a quick way to spot one. Stopping is easy too: once the pairs have met, there is nothing left to find, so you never test past the square root.

• 1 × 36
• 2 × 18
• 3 × 12
• 4 × 9
• 6 × 6

Where the mix-up actually costs marks

The two words carry straight into a piece of examined work that appears on every board: highest common factor and lowest common multiple. Swapping them does not produce a slightly wrong answer, it produces an answer of visibly the wrong size, which is good news - it means the error can be caught before the working is handed in.

For 12 and 18, the HCF is 6, which is smaller than both, and the LCM is 36, which is larger than both. An HCF bigger than either of the original numbers is impossible. An LCM smaller than either of them is impossible. Reading your answer against those two rules takes a second and catches the swap every time.

The questions rarely use the words. They say two lighthouses flash every 12 and 18 seconds, when do they next flash together - which is the LCM, 36 seconds. Or: what is the largest square tile that will fill a 12 cm by 18 cm rectangle exactly - which is the HCF, 6 cm. The size check works on those too, because a tile cannot be wider than the wall.